document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174804>Question 1174804</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the fields medal the highest scientific award for mathematics is awarded evey 4 years. It was awarded in 2010. the birkhoff prize for applied mathematics is awarded evry 3 years. It was awarded in 2009. What are the first two times when both awards were given in the same year in 21st century? </SPAN>\n" );
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document.write( "Edit: I've included the year 2018 as part of the answer after rereading the problem.\r<BR>\n" );
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document.write( "The Fields Medal is awarded every 4 years.<BR>\n" );
document.write( "The Birkhoff Prize is awarded every 3 years.\r<BR>\n" );
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document.write( "With 2010 as a reference point for the Fields Medal, let's write a few numbers that are 4 spaces ahead and behind this reference year.<BR>\n" );
document.write( "{..., 1990, 1994, 1998, 2002, <font color=red>2006</font>, 2010, 2014, <font color=blue>2018</font>, 2022, 2026, 2030, ...}\r<BR>\n" );
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document.write( "Do a similar thing for the Birkhoff Prize years. We start at 2009. Add and subtract multiples of 3 to represent possible years the prize is given<BR>\n" );
document.write( "{..., 1994, 1997, 2000, 2003, <font color=red>2006</font>, 2009, 2012, 2015, <font color=blue>2018</font>, 2021, 2024...}\r<BR>\n" );
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document.write( "The year marked in red is where the two prizes line up, and also the smallest such year in the 21st century (which starts at the year 2000). \r<BR>\n" );
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document.write( "Note how 4*3 = 12 is the LCM of 4 and 3. This tells us that the two awards will line up every 12 years. So 2006+12 = <font color=blue>2018</font> is the next time the two awards will be given together. To help confirm this, the year 2018 is found in both sets listed above. \r<BR>\n" );
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document.write( "Also, note that 2006-12 = 1994 was the last time prior to the 2006 year, is when the two awards lined up. The year 1994 is found in both sets listed above.\r<BR>\n" );
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document.write( "These years are when the two awards lined up<BR>\n" );
document.write( "{..., 1994, <font color=red>2006</font>, <font color=blue>2018</font>, ...}\r<BR>\n" );
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document.write( "Answer: The year <font color=red>2006</font> and <font color=blue>2018</font><BR>\n" );
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